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| In [[mathematics]], and in particular [[functional analysis]], the '''tensor product of [[Hilbert space]]s''' is a way to extend the [[tensor product]] construction so that the result of taking a tensor product of two Hilbert space is another Hilbert space. Roughly speaking, the tensor product is the metric space [[complete metric space|completion]] of the ordinary tensor product. This is a special case of a [[topological tensor product]]. The tensor product allows the Hilbert space to be described by a [[symmetric monoidal category]].<ref>B. Coecke and E. O. Paquette, Categories for the practising physicist, in: New Structures for Physics, B. Coecke (ed.), Springer Lecture Notes in Physics, 2009. [http://arxiv.org/abs/0905.3010 arXiv:0905.3010]</ref>
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| ==Definition==
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| Since Hilbert spaces have [[Dot product|inner products]], one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors. Let ''H''<sub>1</sub> and ''H''<sub>2</sub> be two Hilbert spaces with inner products <math>\langle \cdot,\cdot\rangle_1</math> and <math>\langle \cdot,\cdot\rangle_2</math>, respectively. Construct the tensor product of ''H''<sub>1</sub> and ''H''<sub>2</sub> as vector spaces as explained in the article on [[tensor product]]s. We can turn this vector space tensor product into an [[inner product space]] by defining
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| :<math> \langle\phi_1\otimes\phi_2,\psi_1\otimes\psi_2\rangle = \langle\phi_1,\psi_1\rangle_1 \, \langle\phi_2,\psi_2\rangle_2 \quad \mbox{for all } \phi_1,\psi_1 \in H_1 \mbox{ and } \phi_2,\psi_2 \in H_2 </math> | |
| and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on ''H''<sub>1</sub> × ''H''<sub>2</sub> and linear functionals on their vector space tensor product. Finally, take the [[complete space#Completion|completion]] under this inner product. The resulting Hilbert space is the tensor product of ''H''<sub>1</sub> and ''H''<sub>2</sub>.
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| ===Explicit construction===
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| The tensor product can also be defined without appealing to the metric space completion. If ''H''<sub>1</sub> and ''H''<sub>2</sub> are two Hilbert spaces, one associates to every [[simple tensor]] product <math>x_1 \otimes x_2</math> the rank one operator from ''H''<sub>1</sub><sup>∗</sup> to ''H''<sub>2</sub> that maps a given <math>x^*\in H^*_1</math> as
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| :<math> x^* \mapsto x^*(x_1) \, x_2</math>
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| This extends to a linear identification between <math> H_1 \otimes H_2</math> and the space of finite rank operators from ''H''<sub>1</sub><sup>∗</sup> to ''H''<sub>2</sub>.
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| The finite rank operators are embedded in the Hilbert space ''HS''(''H''<sub>1</sub><sup>∗</sup>, ''H''<sub>2</sub>) of [[Hilbert-Schmidt operator]]s from ''H''<sub>1</sub><sup>∗</sup> to ''H''<sub>2</sub>. The scalar product in ''HS''(''H''<sub>1</sub><sup>∗</sup>, ''H''<sub>2</sub>) is given by
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| :<math> \langle T_1, T_2 \rangle = \sum_n \langle T_1 e_n^*, T_2 e_n^* \rangle, </math>
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| where <math>(e_n^*)</math> is an arbitrary orthonormal basis of ''H''<sub>1</sub><sup>∗</sup>.
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| Under the preceding identification, one can define the Hilbertian tensor product of ''H''<sub>1</sub> and ''H''<sub>2</sub>, that is isometrically and linearly isomorphic to ''HS''(''H''<sub>1</sub><sup>∗</sup>, ''H''<sub>2</sub>).
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| ===Universal property===
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| The Hilbert tensor product <math>H=H_1\otimes H_2</math> is characterized by the following [[universal property]] {{harv|Kadison|Ringrose|1983|loc=Theorem 2.6.4}}:
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| *There is a [[Hilbert-Schmidt operator|weakly Hilbert-Schmidt mapping]] ''p'' : ''H''<sub>1</sub> × ''H''<sub>2</sub> → ''H'' such that, given any weakly Hilbert-Schmidt mapping ''L'' : ''H''<sub>1</sub> × ''H''<sub>2</sub> → ''K'' to a Hilbert space ''K'', there is a unique bounded operator ''T'' : ''H'' → ''K'' such that ''L'' = ''Tp''.
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| As with any universal property, this characterizes the tensor product ''H'' uniquely, up to isomorphism. The same universal property, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces. It is essentially the same universal property shared by all definitions of tensor products, irrespective of the spaces being tensored: this implies that any space with a tensor product is a [[symmetric monoidal category]], and Hilbert spaces are a particular example thereof.
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| ===Infinite tensor products===
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| If <math>H_n</math> is a collection of Hilbert spaces and <math>\xi_n</math> is a collection of unit vectors in these Hilbert spaces then the incomplete tensor product (or Guichardet tensor product) is the <math>L^2</math> completion of the set of all finite linear combinations of simple tensor vectors <math>\otimes_{n=1}^{\infty} \psi_n</math> where all but finitely many of the <math>\psi_n</math>'s equal the corresponding <math>\xi_n</math>.<ref name="BR">Bratteli, O. and Robinson, D: ''Operator Algebras and Quantum Statistical Mechanics v.1, 2nd ed.'', page 144. Springer-Verlag, 2002.</ref>
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| ===Operator algebras===
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| Let <math>\mathfrak{A}_i</math> be the [[von Neumann algebra]] of bounded operators on <math>H_i</math> for <math>i=1,2</math>. Then the von Neumann tensor product of the von Neumann algebras is the strong completion of the set of all finite linear combinations of simple tensor products <math>A_1\otimes A_2</math> where <math>A_i \in \mathfrak{A}_i</math> for <math>i=1,2</math>. This is exactly equal to the von Neumann algebra of bounded operators of <math>H_1\otimes H_2</math>. Unlike for Hilbert spaces, one may take infinite tensor products of von Neumann algebras, and for that matter [[C* algebra|C*-algebras]] of operators, without defining reference states.<ref name="BR"/> This is one advantage of the "algebraic" method in quantum statistical mechanics.
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| ==Properties==
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| If ''H''<sub>1</sub> and ''H''<sub>2</sub> have [[orthonormal basis|orthonormal bases]] {φ<sub>''k''</sub>} and {ψ<sub>''l''</sub>}, respectively, then {φ<sub>''k''</sub> ⊗ ψ<sub>''l''</sub>} is an orthonormal basis for ''H''<sub>1</sub> ⊗ ''H''<sub>2</sub>. In particular, the Hilbert dimension of the tensor product is the product (as [[cardinal number]]s) of the Hilbert dimensions.
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| ==Examples and applications==
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| The following examples show how tensor products arise naturally.
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| Given two [[measure space]]s ''X'' and ''Y'', with measures μ and ν respectively, one may look at [[Lp space|L]]<sup>2</sup>(''X'' × ''Y''), the space of functions on ''X'' × ''Y'' that are square integrable with respect to the product measure μ × ν. If ''f'' is a square integrable function on ''X'', and ''g'' is a square integrable function on ''Y'', then we can define a function ''h'' on ''X'' × ''Y'' by ''h''(''x'',''y'') = ''f''(''x'') ''g''(''y''). The definition of the product measure ensures that all functions of this form are square integrable, so this defines a [[bilinear]]{{dn|date=December 2013}} mapping L<sup>2</sup>(''X'') × L<sup>2</sup>(''Y'') → L<sup>2</sup>(''X'' × ''Y''). [[Linear combination]]s of functions of the form ''f''(''x'') ''g''(''y'') are also in L<sup>2</sup>(''X'' × ''Y''). It turns out that the set of linear combinations is in fact dense in L<sup>2</sup>(''X'' × ''Y''), if L<sup>2</sup>(''X'') and L<sup>2</sup>(''Y'') are separable. This shows that L<sup>2</sup>(''X'') ⊗ L<sup>2</sup>(''Y'') is [[isomorphic]] to L<sup>2</sup>(''X'' × ''Y''), and it also explains why we need to take the completion in the construction of the Hilbert space tensor product.
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| Similarly, we can show that L<sup>2</sup>(''X''; ''H''), denoting the space of square integrable functions ''X'' → ''H'', is isomorphic to L<sup>2</sup>(''X'') ⊗ ''H'' if this space is separable. The isomorphism maps ''f''(''x'') ⊗ φ ∈ L<sup>2</sup>(''X'') ⊗ ''H'' to ''f''(''x'')φ ∈ L<sup>2</sup>(''X''; ''H''). We can combine this with the previous example and conclude that L<sup>2</sup>(''X'') ⊗ L<sup>2</sup>(''Y'') and L<sup>2</sup>(''X'' × ''Y'') are both isomorphic to L<sup>2</sup>(''X''; L<sup>2</sup>(''Y'')).
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| Tensor products of Hilbert spaces arise often in [[quantum mechanics]]. If some particle is described by the Hilbert space ''H''<sub>1</sub>, and another particle is described by ''H''<sub>2</sub>, then the system consisting of both particles is described by the tensor product of ''H''<sub>1</sub> and ''H''<sub>2</sub>. For example, the state space of a [[quantum harmonic oscillator]] is L<sup>2</sup>('''R'''), so the state space of two oscillators is L<sup>2</sup>('''R''') ⊗ L<sup>2</sup>('''R'''), which is isomorphic to L<sup>2</sup>('''R'''<sup>2</sup>). Therefore, the two-particle system is described by wave functions of the form φ(''x''<sub>1</sub>, ''x''<sub>2</sub>). A more intricate example is provided by the [[Fock space]]s, which describe a variable number of particles.
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| ==References==
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| {{reflist}}
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| *{{Citation | last1=Kadison | first1=Richard V. | last2=Ringrose | first2=John R. | title=Fundamentals of the theory of operator algebras. Vol. I | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-0819-1 | id={{MathSciNet | id = 1468229}} | year=1997 | volume=15}}.
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| *{{Citation | last1=Weidmann | first1=Joachim | title=Linear operators in Hilbert spaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90427-6 | id={{MathSciNet | id = 566954}} | year=1980 | volume=68}}.
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| {{Functional Analysis}}
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| [[Category:Functional analysis]]
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| [[Category:Hilbert space]]
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